Column A: The circumference of the circle
Column B: 12
Column A: The circumference of the circle
Column B: 12
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is A
Solution and Explanation
This question requires understanding the properties of circles and triangles. Specifically, we use the relationship between the center of a circle, its radii, and the properties of an equilateral triangle to find the circle's circumference.
Step 2: Key Formula or Approach:
The circumference of a circle is given by the formula \(C = 2\pi r\), where \(r\) is the radius.
The perimeter of a triangle is the sum of its three sides.
Step 3: Detailed Explanation:
In the given circle with center O, OA and OB are radii. Therefore, \(OA = OB = r\).
This means that \(\Delta AOB\) is an isosceles triangle.
In an isosceles triangle, the angles opposite the equal sides are equal. So, \(\angle OAB = \angle OBA\).
The sum of angles in a triangle is \(180^{\circ}\).
\[ \angle AOB + \angle OAB + \angle OBA = 180^{\circ} \] We are given \(\angle AOB = 60^{\circ}\).
\[ 60^{\circ} + \angle OAB + \angle OAB = 180^{\circ} \] \[ 2\angle OAB = 180^{\circ} - 60^{\circ} = 120^{\circ} \] \[ \angle OAB = 60^{\circ} \] Since all three angles of \(\Delta AOB\) are \(60^{\circ}\), it is an equilateral triangle.
In an equilateral triangle, all sides are equal. Therefore, \(OA = OB = AB = r\).
The perimeter of \(\Delta AOB\) is given as 6.
\[ \text{Perimeter} = OA + OB + AB = r + r + r = 3r \] \[ 3r = 6 \] \[ r = \frac{6}{3} = 2 \] Now we can calculate the circumference of the circle.
\[ C = 2\pi r = 2\pi(2) = 4\pi \] Step 4: Comparing the Quantities:
Column A: The circumference of the circle = \(4\pi\).
Column B: 12.
To compare \(4\pi\) and 12, we use the approximation \(\pi \approx 3.14159\).
\[ 4\pi \approx 4 \times 3.14159 = 12.56636 \] Since \(12.56636 \textgreater 12\), the quantity in Column A is greater.
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